Well, to what I understand from your question is that, as a PMO you would want to mentor the project managers. Is that correct?
The general practice is that the project managers should be managed by the Program managers. The program managers should be managed by the Portfolio managers.
The PMO is a strategic unit having dotted line structure reporting to Quality / Program Portfolio Management Office (PPMO).
The PMO is an evolving body that could help at all the layers of the organization suchlike HR, Sales, Administration, Finance, Delivery, etc . For example the office can question and can find the answers for the challenges the business is facing.
To answer your question:
To check on the quality standards: you will need to initiate Project Quality Assurance (PQA) task force - a subset of Software Quality Assurance Team (SQA). This is actually a big story. But to cut short, this team ensures that your project quality and your organization quality is maintained. The PQA need not be a FTE, can be assigned to your existing team members on rotation basis which will have multi-fold benefits.
AP & AR schedule - There should be a tight integration between the PMO Analyst, the accounts team, the sales team and the delivery team. As the delivery team completes the milestones and receives approval from the client, the PMO analyst would mark it complete for invoicing, the accounts team triggers the invoice(s) and submits to the Client for the processing. The sales team follows-up for the closure. I am sure by now you understand why the PMO is considered vital.
Please do not overload your Project Managers, they have better things to do like communicating with stakeholders, estimation, publish reports, managing the deliverables, analyzing risks, identifying and allocating resources and much more.
I can keep writing at length on the PMO aspects and the good practices, as I have implemented full fledged PPMO (end-to-end) but limiting to your context.
If I lay hands on any books on the PMO subject, I will update here.
Should you have any questions on the PMO, please feel free to shoot.